1220018-Ch05_053-064_TG 10/17/07 11:02 AM Page 53 5 Momentum 5-1 Impulse and Momentum Vocabulary Momentum: A measure of how difficult it is to stop a moving object. momentum ϭ (mass)(velocity) or p ϭ mv If the momentum of an object is changing, as it is when a force is exerted to start it or stop it, the change in momentum can be found by looking at the change in mass and velocity during the interval. change in momentum ϭ change in [(mass)(velocity)] or ⌬p ϭ⌬(mv) For all the exercises in this book, assume that the mass of the object remains constant, and consider only the change in velocity, ⌬v, which is equal to Ϫ vf vo. Momentum is a vector quantity. Its direction is in the direction of the object’s velocity. The SI unit for momentum is the kilogramиmeter/second (kgиm/s). Vocabulary Impulse: The product of the force exerted on an object and the time interval during which it acts. impulse ϭ (force)(elapsed time)orJ ϭ F⌬t The SI unit for impulse is the newtonиsecond (Nиs). The impulse given to an object is equal to the change in momentum of the object. F⌬t ϭ m⌬v The same change in momentum may be the result of a large force exerted for a short time, or a small force exerted for a long time. In other words, impulse is the thing that you do, while change in momentum is the thing that you see. The units for impulse and momentum are equivalent. Remember, 1 N ϭ 1 kgиm/s2. Therefore, 1 Nиs ϭ 1 kgиm/s. 53 1220018-Ch05_053-064_TG 10/17/07 11:02 AM Page 54 Solved Examples Example 1: Tiger Woods hits a 0.050-kg golf ball, giving it a speed of 75 m/s. What impulse does he impart to the ball? (Read more about Tiger Woods at http://www.tigerwoods.com) Solution: Because the impulse equals the change in momentum, you can reword this exercise to read, “What was the ball’s change in momentum?” It is understood that the ball was initially at rest, so its initial speed was 0 m/s. Given: m ϭ 0.050 kg Unknown: ⌬p ϭ ? ⌬v ϭ 75 m/s Original equation: ⌬p ϭ m⌬v Solve: ⌬p ϭ (0.050 kg)(75 m/s) ϭ 3.8 kgиm/s Example 2: Wayne hits a stationary 0.12-kg hockey puck with a force that lasts for Ϫ 1.0 ϫ 10 2 s and makes the puck shoot across the ice with a speed of 20.0 m/s, scoring a goal for the team. With what force did Wayne hit the puck? Given: m ϭ 0.12 kg Unknown: F ϭ ? ⌬v ϭ 20.0 m/s Original equation: F⌬t ϭ m⌬v Ϫ ⌬t ϭ 1.0 ϫ 10 2 s ¢ 0.12 kg 20.0 m s ϭϭm v ϭ и 2 ϭ Solve: F 1 21 Ϫ > 2 240 kg m/s 240 N ¢t 1.0 ϫ 10 2 s Example 3: A tennis ball traveling at 10.0 m/s is returned by Venus Williams. It leaves her racket with a speed of 36.0 m/s in the opposite direction from which it came. a) What is the change in momentum of the tennis ball? b) If the 0.060-kg ball is in contact with the racket for 0.020 s, with what average force has Venus hit the ball? Solution: In this exercise, the tennis ball is coming toward Venus with a speed of 10.0 m/s in one direction, but she hits it back with a speed of 36.0 m/s in the opposite direction. Therefore, you must think about velocity vectors and call one direction positive and the opposite direction negative. ϭϪ ⌬ ϭ a. Given: vo 10.0 m/s Unknown: p ? ϭ ⌬ ϭ ⌬ ϭ Ϫ vf 36.0 m/s Original equation: p m v m(vf vo) m ϭ 0.060 kg ⌬ ϭ Ϫ ϭ Ϫ Ϫ ϭ и Solve: p m(vf vo) (0.060 kg)[36.0 m/s ( 10.0 m/s)] 2.8 kg m/s b. Given: m ϭ 0.060 kg Unknown: F ϭ ? ⌬v ϭ 46.0 m/s Original equation: F⌬t ϭ m⌬v ⌬t ϭ 0.020 s m¢v 0.060 kg 46.0 m s Solve: F ϭϭ1 21 > 2 ϭ140 N ¢t 0.020 s 1 2 54 Momentum 1220018-Ch05_053-064_TG 10/17/07 11:02 AM Page 55 Example 4: To demonstrate his new high-speed camera, Flash performs an experiment in the physics lab in which he shoots a pellet gun at a pumpkin to record the moment of impact on film. The 1.0-g pellet travels at 100. m/s until it embeds itself 2.0 cm into the pumpkin. What average force does the pumpkin exert to stop the pellet? Solution: First, convert g to kg and cm to m. 1.0 g ϭ 0.0010 kg 2.0 cm ϭ 0.020 m Before you can solve for the force in the exercise, you must first know how long the force is being exerted. Remember, in order to find the time, you must use the average velocity, vav. ϩ ϩ vf vo 0 m s 100. m s v ϭϭ> > ϭ50.0 m/s av 2 2 Given: v ϭ 50.0 m/s Unknown: ⌬t ϭ ? ⌬d ϭ 0.020 m Original equation: ⌬d ϭ v⌬t ¢d 0.020 m Solve: ⌬t ϭϭ ϭ0.00040 s v 50.0 m s > Now we can solve for the force the pumpkin exerts to stop the pellet. Given: m ϭ 0.0010 kg Unknown: F ϭ ? ⌬v ϭ 100. m/s Original equation: F⌬t ϭ m⌬v ⌬t ϭ 0.0040 s m¢v 0.0010 kg 100. m s Solve: F ϭϭ1 21 > 2 ϭ250 N ¢t 0.00040 s 1 2 Practice Exercises Exercise 1: On April 15, 1912, the luxury cruiseliner Titanic sank after running into an iceberg. a) What momentum would the 4.23 ϫ 108-kg ship have imparted to the iceberg if it had hit the iceberg head-on with a speed of 11.6 m/s? (Actually, the impact was a glancing blow.) b) If the captain of the ship had seen the iceberg a kilometer ahead and had tried to slow down, why would this have been a futile effort? (Read more about the Titanic at http://www.titanic.com) a) p ϭ mv ϭ (4.23 ϫ 108 kg)(11.6 m/s) ϭ 4.91 ϫ 109 kgиm/s b) The ship’s momentum is so great that the ship is very difficult to stop. 4.91 ϫ 109 kgиm/s Answer: a. Answer: b. see above Momentum 55 1220018-Ch05_053-064_TG 11/20/07 3:32 PM Page 56 Exercise 2: Auto companies frequently test the safety of automobiles by putting them through crash tests to observe the integrity of the passenger compartment. If a 1000.-kg car is sent toward a cement wall with a speed of 14 m/s and the Ϫ impact brings it to a stop in 8.00 ϫ 10 2 s, with what average force is it brought to rest? Ϫ F ϭ m⌬v/⌬t ϭ (1000. kg)(14.0 m/s)/(8.00 ϫ 10 2 s) ϭ 175,000 N Answer: 175,000 N Exercise 3: Rhonda, who has a mass of 60.0 kg, is riding at 25.0 m/s in her sports car when she must suddenly slam on the brakes to avoid hitting a dog crossing the road. She is wearing her seatbelt, which brings her body to a stop in 0.400 s. a) What average force did the seatbelt exert on her? b) If she had not been wearing her seatbelt, and the windshield had stopped her head in Ϫ 1.0 ϫ 10 3 s, what average force would the windshield have exerted on her? c) How many times greater is the stopping force of the windshield than the seatbelt? a) F ϭ m⌬v/⌬t ϭ (60.0 kg)(25.0 m/s)/(0.400 s) ϭ 3750 N Ϫ b) F ϭ m⌬v/⌬t ϭ (60.0 kg)(25 m/s)/(1.0 ϫ 10 3 s) ϭ 1.5 ϫ 106 N c) (1.5 ϫ 106 N)/(3750 N) ϭ 400 times as great Answer: a. 3750 N ϫ 6 Answer: b. 1.5 10 N Answer: c. 400 times as great Exercise 4: On October 17, 2006, the population of the United States reached 300. million. If all of those people jumped up in the air simultaneously, pushing Earth with an average force of 800. N each for a time of 0.10 s, what would happen to the 5.98 ϫ 1024 kg Earth? Show a calculation that justifies your answer. Ϫ ⌬v ϭ F⌬t/m ϭ (300. ϫ 106)(800. N)(0.10 s)/(5.98 ϫ 1024 kg) ϭ 4.0 ϫ 10 15 m/s Therefore, Earth’s motion would not be measurable. ϫ Ϫ15 Answer: 4.0 10 m/s 56 Momentum 1220018-Ch05_053-064_TG 10/17/07 11:02 AM Page 57 Exercise 5: In Sharkey’s Billiard Academy, Maurice is waiting to make his last shot. He notices that the cue ball is lined up for a perfect head-on collision, as shown.